Abstract
We consider the construction of tight Gabor frames (h, a=1,
b=1) from Gabor systems (g, a=1, b=1) with g a window
having few zeros in the Zak transform domain via the operation
h=Z
-1(Zg/|Zg|), where Z is the standard Zak transform. We
consider this operation with g the Gaussian, the hyperbolic
secant, and for g belonging to a class of positive, even,
unimodal, rapidly decaying windows of which the two-sided
exponential is a typical example. All these windows g have the
property that Zg has a single zero, viz. at
(1/2,\1/2), in the unit square [0,1)2. The Gaussian
and hyperbolic secant yield a frame for any a,b > 0 with ab < 1,
and we show that so does the two-sided exponential. For these
three windows it holds that S
a
-1/2
g
h as a
1,
where S
a
is the frame operator corresponding to the Gabor frame
(g,a,a). It turns out that the h’s corresponding to g’s of
the above type look and behave quite similarly when scaling
parameters are set appropriately. We give a particular detailed
analysis of the h corresponding to the two-sided exponential. We
give several representations of this h, and we show that
, and is continuous and
differentiable everywhere except at the half-integers, etc., and
we pay particular attention to the cases that the time constant of
the two-sided exponential g tends to
. We also consider
the cases that the time constants of the Gaussian and of the
hyperbolic secant tend to 0 or to
. It so turns out that
h thus obtained changes from the box function
into its Fourier transform
when the time constant changes from 0 to
Similar content being viewed by others
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Hans G. Feichtinger.
Rights and permissions
About this article
Cite this article
Janssen, A. On Generating Tight Gabor Frames at Critical Density. J. Fourier Anal. Appl. 9, 175–214 (2003). https://doi.org/10.1007/s00041-003-0011-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00041-003-0011-3